Download Final Exam Problems - Electromagnetic Fields II | PHYS 436 and more Exams Physics in PDF only on Docsity!
Physics 436 Final Exam December 15, 2007
3 hours. Open notes, closed book. Do all four problems: 10 points each.
- Consider a plane wave in vacuum traveling along
n =
1
2
x +
y
, as shown:
a.
E must be perpendicular to
k. Which of Maxwell’s equations tells us this? Prove it.
b. Write an
E
0
that describes linear polarization along the z - axis. The answer is not unique.
Write any
E
0
that works.
c. Write an
E
0
that describes a polarization in which the electric field rotates around
k
according to the right hand rule ( i.e. , right circular polarization).
d. For the
E
0
in part c, write the corresponding
B
0
- Consider a material in which the relation between an electromagnetic wave’s k and ω is
k =
a
!
, where a is a constant.
a. Calculate the phase and group velocities as functions of frequency.
b. Suppose we have a wave packet that would have a definite wavelength, λ, except that it is
limited to a finite length, L. Estimate the rate at which it spreads out. That is, roughly
what range of velocities is spanned by this wave packet?
(two more problems on the back)
x
y
n
E =
E
0
e
i
!
k!
!
r " # t ( )
- Recall that an EM wave in a material has B
0
n
c
E
0
, where n is the index of refraction.
Because n is a property of the material, it is defined in the reference frame of the material.
Let’s consider a wave moving in the +
x direction that is linearly polarized along
y.
Suppose we observe the material to be moving along
x with velocity v (which may be
positive or negative).
a. What index of refraction, n! , do we measure for the propagation of the wave described
above?
b. Explain in words what happens when v =!
c
n
- To enhance the directionality of electric dipole radiation (wavelength λ), our intrepid
physicist places the source a distance d away from a conducting plane, as shown. The dipole
oscillates along z ˆ. The intensity distribution would be A sin
2
! in the absence of the
conductor. (I don’t want you to worry about all the constants.)
a. For a given λ, what value of d will maximize the intensity of radiation in the
y direction?
There are multiple solutions; I want the smallest one.
b. What is the angular distribution of the radiation intensity? Use spherical coordinates in
which z is the polar axis and f = 0 along
x.
z
d
p
x
y
